Finite Group Databasebeta

← #309109

Subgroups — #309109

Order 512.

572 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic, complemented
2#26533C211yesmaximal, characteristic
3#26527C211yesmaximal, characteristic, complemented
4#23216C211yesmaximal, characteristic
5#10596C211yesmaximal, characteristic, complemented
6#10595C211yesmaximal, characteristic
7#10353C211yesmaximal, characteristic, complemented
8#10351C211yesmaximal, characteristic
9#9830C211yesmaximal, characteristic
10#9738C211yesmaximal, characteristic
11#9615C211yesmaximal, characteristic, complemented
12#9598C211yesmaximal, characteristic, complemented
13#8925C211yesmaximal, characteristic, complemented
14#8924C211yesmaximal, characteristic
15#8284C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17#2571C2211yescharacteristic
18#2571C2211yescharacteristic
19#2366C2211yescharacteristic
20#2365C2211yescharacteristic
21#2097—21no—
22#2092—21no—
23#2085—21no—
24#2085—21no—
25#2066—21no—
26#2065—21no—
27#1990C2211yescharacteristic
28#1988C2211yescharacteristic
29#1982C2211yescharacteristic
30#1982C2211yescharacteristic
31#1981C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1956C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1948C2211yescharacteristic
39#1942—21no—
40#1939C2211yescharacteristic
41#1939C2211yescharacteristic
42#1934—21no—
43#1925—21no—
44#1863—21no—
45#1862—21no—
46#1862—21no—
47#1846—21no—
48#1845C2211yescharacteristic
49#1805—21no—
50#1804—21no—
51C4×#107C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102—21no—
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×C4×Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72#486—22no—
73#486Q814yes—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107D411yescharacteristic
77C2×#107D411yescharacteristic
78C2×#107D411yescharacteristic
79C2×#107D411yescharacteristic
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—41no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#400—41no—
109#400—41no—
110#391—21no—
111#390—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#387—41no—
119#386—41no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382—21no—
123#382—21no—
124#382—21no—
125#382—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—41no—
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#335—22no—
147#334—22no—
148#332—22no—
149#331—22no—
150#322—22no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C23×C4—41no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C2×Q812yes—
169#127—21no—
170#127—21no—
171#127—21no—
172#127C2×Q812yes—
173C4×Q8—42no—
174C4×Q8—81no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—81no—
178C4×D4—81no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—41no—
206C2×C42C2411yescharacteristic
207#107—22no—
208#107—22no—
209#107—41no—
210#107—41no—
211#107—42no—
212#107—81no—
213#106—81no—
214#106—81no—
215#106—81no—
216#106—81no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—81no—
225C4⋊C8—81no—
226C4⋊C8#4112yes—
227#103—42no—
228#103—81no—
229#103—81no—
230#103—81no—
231#102—24no—
232#102—24no—
233#102—42no—
234#102—42no—
235#102—81no—
236#102—81no—
237#102—81no—
238#102—81no—
239#102—82no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—42no—
245C4×C8—81no—
246C4×C8C2×Q812yes—
247C4×C8#4112yes—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260C24—41no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—161no—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8C22⋊Q812yes—
325C2×C8C22⋊Q812yes—
326C2×C8#13512yes—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—82no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—161no—
358C4⋊C4—161no—
359C4⋊C4#13512yes—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—161no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—82no—
372C42C22×Q811yescharacteristic
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C422−1+411yescharacteristic
376C23—42no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23D4211yescharacteristic
384Q8—42no—
385Q8—42no—
386Q8—81no—
387Q8—81no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—81no—
394D4—81no—
395D4—81no—
396D4—81no—
397D4—161no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—164no—
434C2×C4C2×C22⋊Q811yescharacteristic
435C2×C4#53211yescharacteristic
436C2×C4#54211yescharacteristic
437C2×C4#54811yescharacteristic
438C2×C4D4×Q811yescharacteristic
439C2×C4#55411yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4#57511yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C8—22no—
449C8—22no—
450C8—22no—
451C8—42no—
452C8—41no—
453C8—81no—
454C8—81no—
455C8—81no—
456C8—81no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22#245211yescharacteristic
465C22#326711yescharacteristic
466C22#327111yescharacteristic
467C22#328011yescharacteristic
468C22#328711yescharacteristic
469C22#328811yescharacteristic
470C22#328911yescharacteristic
471C4—21no—
472C4—21no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—82no—
480C4—81no—
481C4—81no—
482C4—161no—
483C4—161no—
484C4#308011yescharacteristic
485C4#308411yescharacteristic
486C4#316811yescharacteristic
487C4#317611yescharacteristic
488C4#337111yescharacteristic
489C4#337611yescharacteristic
490C4#338011yescharacteristic
491C4#338011yescharacteristic
492C2—162no—
493C2#1977511yescharacteristic
494C2#1978611yescharacteristic
495C2#1987011yescharacteristic
496C2#1987411yescharacteristic
497C2#2647911yescharacteristic
498C2#2648211yescharacteristic
499C2#2650311yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #28828C2semidirect
3. #26527C2semidirect
5. #10596C2semidirect
7. #10353C2semidirect
11. #9615C2semidirect
12. #9598C2semidirect
13. #8925C2semidirect
15. #8284C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#288282
#265332
#265272
#232162
#105962
#105952
#103532
#103512
#98302
#97382
#96152
#95982
#89252
#89242
#82842